Sketch for the curve `y = sin^(-1)( (2x)/( 1 + x^2))`

Answer

To sketch the curve of \( y = \sin^{-1}\left(\frac{2x}{1 + x^2}\right) \), we will follow a systematic approach.

### Step-by-Step Solution:

1. **Substitution**: Let \( x = \tan(\theta) \). This substitution helps us to simplify the expression. As \( \theta \) varies from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \), \( x \) will cover all real numbers.

\[ \text{Thus, } \frac{2x}{1 + x^2} = \frac{2\tan(\theta)}{1 + \tan^2(\theta)} \]

2. **Using Trigonometric Identity**: We know that \( 1 + \tan^2(\theta) = \sec^2(\theta) \). Therefore, we can simplify:

\[ \frac{2\tan(\theta)}{1 + \tan^2(\theta)} = \frac{2\tan(\theta)}{\sec^2(\theta)} = 2\tan(\theta)\cos^2(\theta) = 2\sin(\theta)\cos(\theta) = \sin(2\theta) \]

3. **Rewriting the Function**: Now we can rewrite the function \( y \):

\[ y = \sin^{-1}(\sin(2\theta)) \]

4. **Understanding the Inverse Function**: The function \( \sin^{-1}(\sin(x)) \) has specific intervals:
- If \( 2\theta \) is in the range \( -\frac{\pi}{2} \leq 2\theta \leq \frac{\pi}{2} \), then \( y = 2\theta \).
- If \( 2\theta \) is in the range \( \frac{\pi}{2} < 2\theta < \frac{3\pi}{2} \), then \( y = \pi - 2\theta \).
- If \( 2\theta \) is in the range \( -\frac{3\pi}{2} < 2\theta < -\frac{\pi}{2} \), then \( y = -\pi - 2\theta \).

5. **Finding the Intervals for \( x \)**: Since \( \theta = \tan^{-1}(x) \), we can express the intervals for \( x \):
- For \( -\frac{\pi}{4} \leq \theta \leq \frac{\pi}{4} \) (which corresponds to \( -1 \leq x \leq 1 \)), we have \( y = 2\tan^{-1}(x) \).
- For \( -\frac{\pi}{4} < \theta < -\frac{3\pi}{4} \) (which corresponds to \( x < -1 \)), we have \( y = -2\tan^{-1}(x) - \pi \).
- For \( \frac{\pi}{4} < \theta < \frac{3\pi}{4} \) (which corresponds to \( x > 1 \)), we have \( y = -2\tan^{-1}(x) + \pi \).

6. **Sketching the Graph**: Now we can sketch the graph based on these intervals:
- For \( -1 \leq x \leq 1 \), plot \( y = 2\tan^{-1}(x) \).
- For \( x < -1 \), plot \( y = -2\tan^{-1}(x) - \pi \).
- For \( x > 1 \), plot \( y = -2\tan^{-1}(x) + \pi \).